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Open AccessArticle

Some New Fractional-Calculus Connections between Mittag–Leffler Functions

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta 99628, TRNC, Mersin-10, Turkey
Department of Mathematics, Cankaya University, Balgat, Ankara 06530, Turkey
Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 485;
Received: 30 March 2019 / Revised: 10 May 2019 / Accepted: 20 May 2019 / Published: 28 May 2019
We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fractional derivative of the two-parameter Mittag–Leffler function, which is in turn a fractional integral of the one-parameter Mittag–Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag–Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse. View Full-Text
Keywords: fractional integrals; fractional derivatives; Mittag–Leffler functions fractional integrals; fractional derivatives; Mittag–Leffler functions
MDPI and ACS Style

Srivastava, H.M.; Fernandez, A.; Baleanu, D. Some New Fractional-Calculus Connections between Mittag–Leffler Functions. Mathematics 2019, 7, 485.

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